ICSE Class X Math 2014 Solved Paper

Question : 28 of 52

Marks: +1, -0

Find the value of '

a

' for which the following points

A(a, 3), B(2,1)

and

C(5, a)

are collinear. Hence find the equation of the line.

Solution:

Given :

{\A}(a, 3), {\B}(2,1)

and

{\C}(5, a)

are collinear.

\; ∴ \;\;\text " Slope of "\; {\AB}\;=\;\text " Slope of "\; {\BC}

⇒\;\;{1-3}/{2-a}\;=\;{a-1}/{5-2}

⇒\;\;{-2}/{2-a}\;=\;{a-1}/{3}

⇒ -6\;=(2-a)(a-1)

⇒ -6\;=2 a-2-a^2+a

⇒\;a^2-3 a-4\;=0

⇒ a^2-4 a+a-4\;=0

⇒\;(a-4)(a+1)\;=0

\;a\;=4,-1

Now equation of straight line are :
when

a=4

\;y-3=\;{1-3}/{2-4}(x-4)

\;y-3=\;{-2}/{-2}(x-4)

y-3\;=x-4

y\;=x-1

when

a=-1

y-3\;=\;{1-3}/{2+1}(x+1)

y-3\;=-\;{2}/{3}(x+1)

3 y-9\;=-2 x-2

3 y+2 x\;=7

Hence the equations of lines are

y=x-1 \;\text " and "\; 3 y+2 x=7

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